Five cards are dealt from a standard deck of 52 cards. What is the probability that three or four of the cards are aces?
Prob(3 or 4 are aces)
= ( C(4,3)C(48,2) + C(4,4)C(48,1) )/ C(52,5)
= ....
To find the probability that three or four of the cards are aces, we need to calculate the number of favorable outcomes and the total number of possible outcomes.
First, let's determine the total number of possible outcomes. When five cards are dealt from a standard deck of 52 cards, the number of possible outcomes can be calculated using combinations. We can use the formula:
nCr = n! / (r!(n - r)!)
Where n is the total number of items to choose from (52 cards) and r is the number of items we are selecting (5 cards). So, the total number of possible outcomes is:
52C5 = 52! / (5!(52 - 5)!)
= 52! / (5!47!)
= (52 * 51 * 50 * 49 * 48) / (5 * 4 * 3 * 2 * 1)
= 2,598,960
Now, let's determine the number of favorable outcomes. We have two scenarios to consider: three aces and four aces.
For three aces:
To calculate the number of favorable outcomes, we need to select 3 aces out of the 4 available aces and 2 non-ace cards out of the remaining 48 non-ace cards.
Number of favorable outcomes = (4C3) * (48C2)
= 4 * (48! / (2!(48 - 2)!)
= 4 * (48! / (2!46!)
= 4 * (48 * 47) / (2 * 1)
= 4 * 1128
= 4512
For four aces:
To calculate the number of favorable outcomes, we need to select 4 aces out of the 4 available aces and 1 non-ace card out of the remaining 48 non-ace cards.
Number of favorable outcomes = (4C4) * (48C1)
= 1 * (48! / (1!(48 - 1)!)
= 1 * (48! / (1!47!)
= 1 * 48
= 48
Now, we can calculate the total number of favorable outcomes by summing the number of favorable outcomes for each scenario:
Total number of favorable outcomes = Number of favorable outcomes for three aces + Number of favorable outcomes for four aces
= 4512 + 48
= 4560
Therefore, the probability that three or four of the cards are aces is:
Probability = Total number of favorable outcomes / Total number of possible outcomes
= 4560 / 2,598,960
≈ 0.00176, or 0.176%
So, the probability is approximately 0.00176 or 0.176%.